Seebeck effect of Dirac electrons

We study the Seebeck effect in the three-dimensional Dirac electron system based on the linear response theory with Luttingers gravitational potential. The Seebeck coefficient $S$ is defined by $S = L_{12} / L_{11} T$, where $T$ is the temperature, and $L_{11}$ and $L_{12}$ are the longitudinal response coefficients of the charge current to the electric field and to the temperature gradient, respectively; $L_{11}$ is the electric conductivity and $L_{12}$ is the thermo-electric conductivity. It is confirmed that $L_{11}$ and $L_{12}$ are related through Motts formula in low temperatures. The dependences of the Seebeck coefficient on the chemical potential $mu$ and the temperature $T$ when the chemical potential lies in the band gap ($|mu| < Delta$) are partially captured by $S propto (Delta - mu) / k_{mathrm{B}} T$ for $mu > 0$ as in semiconductors. The Seebeck coefficient takes the relatively large value $|S| simeq 1.7 ,mathrm{m V/K}$ at $T simeq 8.7,mathrm{K}$ for $Delta = 15 ,mathrm{m eV}$ by assuming doped bismuth.